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On 2-absorbing ideals in commutative semirings. (English) Zbl 1343.16042

Summary: We study 2-absorbing ideals in commutative semirings \(S\) with \(1\neq 0\) and prove some important results analogous to ring theory. More general form of the Prime Avoidance Theorem is also given. We also prove that if \(I=\langle a_1,a_2,\ldots,a_r\rangle\) is a finitely generated ideal of a semiring \(S\) and \(P_1,P_2,\ldots,P_n\) are substractive prime ideals of \(S\) such that \(I\nsubseteq P_i\) for each \(1\leqslant i\leqslant n\), then there exist \(b_2,\ldots,b_r\in S\) such that \(c=a_1+b_2a_2+\cdots+b_ra_r\notin \bigcup_{i=1}^nP_i\).

MSC:

16Y60 Semirings
16D25 Ideals in associative algebras